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We show that the diameter of the image of the skinning map on the deformation space of an acylindrical reflection group is bounded by a constant depending only on the topological complexity of the components of its boundary, answering a conjecture of Minsky in the reflection group setting. This result can be interpreted as a uniform rigidity theorem for disk patterns. Our method also establishes a connection between the diameter of the skinning image and certain discrete extremal width on the Coxeter graph of the reflection group.
Submitted 19 August, 2024; originally announced August 2024.
We prove that any degree d rational map having a parabolic fixed point of multiplier 1 with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group Hd+1, with the mating realized by an algebraic correspondence. This solves the parabolic version of the Bullett-Freiberger Conjecture from 2003 on mateability between rational maps and Hecke groups. The proof is in two steps. The first is the construction of a pinched polynomial-like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial-like map to an algebraic correspondence via a suitable branched covering.
Submitted 20 July, 2024; originally announced July 2024.
In this paper, we study matings of (anti-)polynomials and Fuchsian, reflection groups as Schwarz reflections, B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces. We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups. These matings emerge naturally as degenerate (anti-)polynomial-like maps, and we show that the corresponding parameter space slices for such matings bear strong resemblance with parameter spaces of polynomial maps. Furthermore, we provide algebraic descriptions for these matings, and construct algebraic correspondences that combine generic (anti-)polynomials and genus zero orbifolds in a common dynamical plane, providing a new concrete evidence to Fatou's vision of a unified theory of groups and maps.
Submitted 31 July, 2024; originally announced August 2024.
This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is quasiconformally different from the gasket limit set of a geometrically finite Kleinian group.
Submitted 19 February, 2024; originally announced February 2024.
We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become Hénon-like, and then converge super-exponentially fast to the space of one-dimensional unimodal maps. We also completely characterize the local geometry of every stable and… ▽ More
On a Riemann surface, periods of a meromorphic differential along closed loops define a period character from the absolute homology group into the additive group of complex numbers. Fixing the period character in strata of meromorphic differentials defines the isoperiodic foliation where the remaining degrees of freedom are the relative periods between the zeroes of the differential. In strata of meromorphic differentials with exactly two zeroes, leaves have a natural structure of translation surface. In this paper, we give a complete description of the isoperiodic leaves in marked stratum H(1,1,−2) of meromorphic 1-forms with two simple zeroes and a pole of order two on an elliptic curve. For each character, the corresponding leaf is a connected Loch Ness Monster. The translation structures of generic leaves feature a ramified cover of infinite degree over the flat torus defined by the lattice of absolute periods. By comparison, isoperiodic leaves of the unmarked stratum are complex disks endowed with a half-translation structure having infinitely many singular points. Finally, we give a description of the large-scale conformal geometry of the wall-and-chamber decomposition of the leaves.
Submitted 24 January, 2024; v1 submitted 11 May, 2023; originally announced May 2023.
A twisted rational map over a non-archimedean field K is the composition of a rational function over K and a continuous automorphism of K. We explore the dynamics of some twisted rational maps on the Berkovich projective line.
For a hyperbolic polynomial automorphism of C^2 with a disconnected Julia set, and under a mild dissipativity condition, we give a topological description of the components of the Julia set. Namely, there are finitely many "quasi-solenoids" that govern the asymptotic behavior of the orbits of all non-trivial components. This can be viewed as a refined Spectral Decomposition for a hyperbolic map, as well as a two-dimensional version of the (generalized) Branner-Hubbard theory in one-dimensional polynomial dynamics. An important geometric ingredient of the theory is a John-like property of the Julia set in the unstable leaves.
The goal of this survey is to present intimate interactions between four branches of conformal dynamics: iterations of anti-rational maps, actions of Kleinian reflection groups, dynamics generated by Schwarz reflections in quadrature domains, and algebraic correspondences. We start with several examples of Schwarz reflections as well as algebraic correspondences obtained by matings between anti-rational maps and reflection groups, and examples of Julia set realizations for limit sets of reflection groups (including classical Apollonian-like gaskets). We follow up these examples with dynamical relations between explicit Schwarz reflection parameter spaces and parameter spaces of anti-rational maps and of reflection groups. These are complemented by a number of general results and illustrations of important technical tools, such as David surgery and straightening techniques. We also collect several analytic applications of the above theory
We prove {\em a priori} bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials fc:z↦z2+c of bounded type. It implies local connectivity of the corresponding Julia sets J(fc) and MLC (local connectivity of the Mandelbrot set $\Mandel$) at the corresponding parameters c. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg-Khanin-Sinai in the early 1980s.